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The mathematical concept of the Navier-Stokes equations offers nonetheless basic open questions that signify as many demanding situations for the mathematicians. This quantity collects a chain of articles whose aim is to provide new contributions and concepts to those questions, with specific regard to turbulence modelling, regularity of suggestions to the initial-value challenge, movement in zone with an unbounded boundary and compressible circulate.

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Our opening example indicates that molecular interaction between the solid wall with zero tangential velocity (or zero average velocity on the molecular level) should impart the same condition to immediately adjacent fluid. A crucial feature of the new boundary condition ~ =0 it is that it provides a mechanism by which a boundary can produce vorticity in the fluid. We shall describe this in some detail later on. Next we shall discuss some scaling properties of the NavierStokes equations with the aim of introducing a parameter (the Reynolds number) that measures the effect of viscosity on the flow.

It end in a nice regular way on a surface? 2-7? 2-7 Can this be a vortex tube generated by assumption. S7 In particular, note that the assertion often made ,~ that a vortex line cannot end in the fluid is clearly false if we allow to have zeros and probably is false even if ~ ~ has no zeros (an orbit of a vector field can wander around forever without accumulating at an endpoint ... as with a line with irrational slope on a torus). Thus our assertion about vortex tubes "ending" is correct if we interpret "ending" properly.

J. Math. 25 (1973), 1183-1215. 49 au + uau + vau + at ax ay au Waz The change of variables produces a(u'u) at' + U ,a(u'u) ax' + uv'a(u'u) ay' + U ,a(u'u) aw' at' ~ u ax' ay' ay w az' 3"Z" ax- 2 ( U ) (au' + L aP" a2 u ') a;'2 . '. ' at where p = - grad components. 5) are the Navier-Stokes equations in dimensionless variables. d the other with the same fluid but with U00 10000 km/hour and radius ;: 1 meter. be the radius and U to be the velocity Uoc Reynolds number is the same for each flow. If we choose L to at infinity, then the The equations satisfied by the dimensionless variables are thus identical for the two flows.